Course material for M531-532, Jochen Denzler

I do not adopt a textbook as `official' but will mainly use Chicone and Walter. More

Syllabus for M531: (various file formats) syllabus.dvi syllabus.ps syllabus.pdf
Syllabus for M532: (various file formats) syllabus2.dvi syllabus2.ps syllabus2.pdf

Homework Problems:
Homework Problems from M531: Hwk.dvi Hwk.ps Hwk.pdf
Homework Problems for M532: These files will be continuously updated (last update Apr 01).
Hwk2.dvi Hwk2.ps Hwk2.pdf

Upon request, here is a hands-on example on the Jordan Normal Form:
JNFexamples.dvi JNFexamples.ps JNFexamples.pdf

For precise graphics, you find here files that print a decimal grid on a sheet of paper, with the small square 1mm, 2mm or 3mm sidelength. I tested the postscript version of the files on one laser printer. The quality of the actual outcome may be hardware dependent.
size PostScript PDF
1mm 1mm:ps 1mm:pdf
2mm 2mm:ps 2mm:pdf
3mm 3mm:ps 3mm:pdf

Class diary and homework assignments:

Wed Jan 12: Focused review of 531.
Fri Jan 14: Fundamental matrix and Wronskian of systems. Hwk 36 due Wed; 37,40,39 due Fri (ask questions Wed)
Mon Jan 17: MLK DAY
Wed Jan 19: Intermezzo: stability proof (531 leftover)
Fri Jan 21: stability proof finished up.
Mon Jan 24: Proofs re Wronskian; what is D det (A) ?
Wed Jan 26: a bit more on Ddet; back to const. coeffs: linear algebra needed to study invariant subspaces
Fri Jan 28: Question on notation and analysis concepts; what actually is Dexp, Dexp(A), Dexp(A)B (b/c of Hwk 38)
Mon Jan 31: invariant spaces for a linear system; invariant manifolds from the linearization Hwk thru 41; and 44 due Wed
Wed Feb 02: invariant manifolds from the linearization Hwk 45,46 due Monday; Hwk 42, 43, 47,48 Wed
Fri Feb 04: Discussion of examples for IM. (Hwk 45 consumed for discussion); classification of 2D linear phase portraits.
Mon Feb 07: Variation of Parameters: formula, use in a `nonlinearity -> inhomogeneity' philosophy, convolution and supersposition of pulses
Wed Feb 09: Superposition of pulses cont'd Hwk 50,52 due next week
Fri Feb 11: Power Series: Intro.
Mon Feb 14: Discussion; quick-overview: complex differentiability
Wed Feb 16: quick-overview: analytic <=> holomorphic;
Fri Feb 18: Example for a recursion formula used to estimate existence time through power series.
Mon Feb 21: Thm on removable singularities of analytic functions stated. Def of ordinary, regular singular, irregular singular points of a linear ODE with analytic coefficients.
Wed Feb 23: Q&A, mainly regarding hwk 54 (cplx vs real derivative)
Fri Feb 25:
Mon Feb 28 : ordinary / reg'singular / irreg'singular points at infinity
Wed Mar 02: IVP for singular pts not well-defined; Euler ODE's
Fri Mar 04: Hwk 54, da capo
Mon Mar 07: Questions and Answers
Wed Mar 09: FIRST EXAM
Fri Mar 11: How are power series expansions at regular singular points expected to look?
Mon Mar 14: The Airy ODE Hwk: read skipped part of notes on Airy ODE, and do Pblm 59
Wed Mar 16: (double size) (1) Complex logarithm functions, via branch cuts (2) The Bessel ODE, the function Jnu Hwk: 60-63
Fri Mar 18: cancelled; (make-up preceding Wed)
Mon Mar 21: SPRING BREAK
Wed Mar 23: SPRING BREAK
Fri Mar 25: SPRING BREAK (GOOD FRIDAY)
Mon Mar 28: The `big' complex logarithm unifying all braches, defined on a `Riemann surface'
Wed Mar 30: cancelled; (make-up following Fri)
Fri Apr 01: (double size) Logarithmic solutions of the Bessel eqn for integer parameter. Overview over the hypergeometric DE: the significance of having exactly 3 regular singularities and how to reduce any 2d order ODE with exactly 3 regular singularities (and no irregular ones) to the hypergeometric. Hwk: 64-66
Mon Apr 04: Thm about power expansion near regular singular points, and its proof
Wed Apr 06: dito, continued. In particular monodromy. a bunch of hwk is still pending
Fri Apr 08: finding a logarithm of a matrix.
Mon Apr 11: Q&A; growth estimate of solution near a regular singular point; Intro BVP
Wed Apr 13:
Fri Apr 15: EXAM2
Mon Apr 18:
Wed Apr 20:
Fri Apr 22:
Mon Apr 25:
Wed Apr 27:
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Fri Apr 29: STUDY DAY: regular class time will be extra office hour
Thu May 05: FINAL EXAM: 10:15-12:15 (scheduled by university policy)